Flory-Huggins approach

The well-known Flory treatment [50-52] of the en-thropic contribution to the Gibbs energy of mixing of polymers with solvents is still the simplest and most reliable theory developed. It is quite apparent, however, that the Flory-Huggins theory was established on the basis of the experimental behavior of only a few mixtures investigated over a very narrow range of temperature. Strict applications of the Flory-Huggins approach... 

This involves the Flory-Huggins parameter x and hence assumes the same limitation as the rest of the Flory-Huggins approach, i.e. a moderately concentrated solution. Flory and Krigbaum rewrote this equation in terms of some other parameters, i.e. 

Flory-Huggins Approach. One explanation of blend behavior lies in the thermodynamics of the preceding section, where instead of a polymer-solvent mixture, we now have a polymer-polymer mixture. In these instances, the heat of mixing for polymer pairs (labeled 1 and 2) tends to be endothermic and can be approximated using the solubility parameter. The interaction parameter for a polymer-polymer mixture, Xi2, can be approximated by... 

Either of these expressions could be substituted for the ideal partial molal entropy of mixing, as deLned in Equation 2.2. The Flory-Huggins approach, for example, has been used to improve the prediction of the solubility of sulfamethoxypyridazine (Bustamante et al., 1989) and temazepam (Richardson et al., 1992) in a wide variety of solvents, including water. 

Flory-Huggins approach, may be somewhat low even with highly crystalline preparations of A- or B-type starch lintners (DP —15). Despite its theoretical limitations, the Flory analysis (as illustrated in Figure 8.10 inset) can be used to simulate the melting behavior of starch in practical applications (e.g. extrusion cooking, baking) and to compare the thermal stability of different starch materials under dynamic heating at various moisture conditions.20,25 240 337... 

Equation (5.14) represents the link between the specific retention volumes 7 , calculated from chromatographic data, and the thermodynamic interaction parameter, as defined by the Flory-Huggins approach. This parameter depends on the polymer concentration, as has been proved experimentally [12, 13]. Orofino and Flory [14] have shown that the term Xi2 92 iu eqn (5.8) represents an incomplete expression for the non-configurational contribution to the activity, this must be written as a series of powers in terms of the volume fraction, the first term being Xi2[Pg.131]

Note that in the Flory-Huggins approach [eq. (HI.S)] the coefiGcient w is fixed (iv = a ) while it remains as a free parameter in eq. (in.9). The second formulation is slightly more general, but it is restricted to small 4> values. The lattice model has the advantage of providing a description at all d> values. 

The Flory-Huggins lattice consideration of the polyelectrolyte solutions presented above incorrectly describes dilute polyelertrolyte solutions. In the Flory-Huggins approach, the monomers are uniformly distributed over the whole volrrme of the system, leading to underestimation of the effect of the short-range monomer-monomer interactions and of the intra-chain electrostatic interactions. A similar problem appears in the Flory-Huggins theory of phase separation of polymer solutions (see for discussion References 32 and 33). This leads to the incorrect expression for the low polymer density branch of the phase diagram. 

We shall in section 4.2 deal with regular solutions of small-molecule substances. The construction of phase diagrams from the derived equations is demonstrated. The Flory—Huggins mean-field theory derived for mixtures of polymers and small-molecule solvents is dealt with in section 4.3. It turns out that the simple Flory—Huggins theory is inadequate in many cases. The scaling laws for dilute and semi-dilute solutions are briefly presented. The inadequacy of the Flory-Huggins approach has led to the development of the equation-of-state theories which is the fourth topic (section 4.6) Polymer-polymer mixtures are particularly complex and they are dealt with in section 4.7. 

There have been sophisticated calculations of x these systems. In the Flory-Huggins model x is independent of molecular weight and composition. In reality, this parameter has been shown, by scattering experiments and by theoretical calculations to be a function of Af, T and 0. 3,40,49 Theoretical attempts, which are beyond the scope of the mean field predictions of the Flory-Huggins approach which applies stri ct to incompressible systems, have been made to address these questions. 3,4o 49 The theories of Bates and Muthukumar 3 and of Schweizer and Curro °> both have predictions which may be written in the following form... 

The Flory-Huggins approach is not directly capable of predicting lest behavior unless a temperature dependent Xu value exhibiting increasing values (negative to positive) with increasing temperature is employed. The temperature dependence of Xn has often been expressed by Xn = a + (b/T). For polymer-solvent mixtures, Xu has been expressed as a function of both temperature and concentration Xn = a + b/T) + c(j>i + d(j> [10]. 

Biros, et al. [47] compared the predictions of both the equation of state and modified solubility parameter approaches for the dependence of the binary interaction parameter on temperature, pressure, solvent chain length, and polymer flexibility. Both treatments gave qualitatively similar predictions with values of x based on solubility parameters always lower than the equatiiS of state predictions. Quantitative agreement between the two approaches could be achieved if an entropic correction term, represented by a in Eq. (31), were employed. Thus, this modified Flory-Huggins approach should provide a good representation of the equilibrium thermodynamic state of the blends. 

The Gaussian random coil model is appropriate when the scale of inhomogeneity (e.g., the interfacial thickness) is large compared with the length of a bond, b, and the range of interactiOTis, xj/. To handle the case where this is not true, a lattice model has been proposed by Helfand [202 205], in the spirit of the Flory-Huggins approach [206], For infinite molecular weights, he obtained ... 

FsiteVtot/a, and nm=Flory-Huggins approach (eq.2-24), we get ... 

The simple Flory-Huggins approach and the solubility parameter concept are inadequate when tested against experimental data for polymer solutions. Even for mixtures of n-alkanes, the excess thermodynamic properties cannot be described satisfactorily - Flory et In particular, changes of volume upon mixing are excluded and observed... 

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